The null result of the celebrated 1887 Michelson–Morleyexperiment was surprising and difficult to explain in

terms of then prevalent physics concepts. It required a fun-damental change in the notions of space and time and wasfinally explained, almost 20 years later, by Albert Ein-stein’s special theory of relativity. (See the May 1987 spe-cial issue of PHYSICS TODAY devoted to the centennial ofthe experiment.) Special relativity postulates that all lawsof physics are invariant under Lorentz transformations,which include ordinary rotations and changes in the ve-locity of a reference frame. Subsequently, quantum fieldtheories all incorporated Lorentz invariance in their basicstructure. General relativity includes the invariancethrough Einstein’s equivalence principle, which impliesthat any experiment conducted in a small, freely fallinglaboratory is invariant under Lorentz transformations.That result is known as local Lorentz invariance.

Experimental techniques introduced throughout the20th century led to continued improvements in tests ofspecial relativity. For example, 25 years ago, Alain Brilletand John L. Hall used a helium–neon laser mounted on arotary platform to improve the accuracy of the Michel-son–Morley experiment by a factor of 4000. In addition tothe Michelson–Morley experiments that look for ananisotropy in the speed of light, two other types of exper-iments have constrained deviations from special relativity.Kennedy–Thorndike experiments search for a dependenceof the speed of light on the lab’s velocity relative to a pre-ferred frame, and Ives–Stilwell experiments test specialrelativistic time dilation.

In 1960, Vernon Hughes and coworkers and, inde-pendently, Ron Drever conducted a different kind ofLorentz invariance test.1 They measured the nuclear spinprecession frequency in lithium-7 and looked for changesin frequency or linewidth as the direction of the magneticfield rotated, together with Earth, relative to a galactic ref-erence frame. Such measurements, known as Hughes–Drever experiments, have been interpreted, for example,in terms of a possible difference between the speed of lightand the limiting velocity of massive particles.2

Why bother?Theorists and experimentalists indisciplines ranging from atomicphysics to cosmology have been in-creasingly interested in tests ofLorentz invariance. The high sensi-tivity of experimental tests combinedwith recent advances in their theo-retical interpretation allows one to

probe ultrashort distance scales well beyond the reach ofconventional particle-collider experiments. In fact, boththe best experiments and astrophysical observations canindirectly probe distance scales as short as the Plancklength LPl ⊂ (G\/c3)1/2 � 10⊗35 m. Experiments that probesuch short scales can constrain quantum gravity scenarios.

The breaking of Lorentz symmetry enables the CPTsymmetry, which combines charge conjugation (C), parity(P), and time-reversal (T) symmetries, to be violated. Inconventional field theories, the Lorentz and CPT symme-tries are automatically preserved. But in quantum grav-ity, certain restrictive conditions such as locality may nolonger hold, and the symmetries may be broken. Thebreaking of CPT, combined with baryon-number violation,could be the source of the dynamically generated domi-nance of matter over antimatter in the universe. Unlike amore conventional scenario involving only CP violation,baryogenesis based on CPT violation would not require adeparture from thermal equilibrium. (See the article byHelen Quinn, PHYSICS TODAY, February 2003, page 30.)

Cosmology provides an additional important impetusto look for violations of Lorentz symmetry. The recognitionthat the universe is dominated by dark energy suggests anew field—known as quintessence—that permeates allspace. The interaction of that field with matter would man-ifest itself as an apparent breaking of Lorentz symmetry.

It could be argued on aesthetic grounds that theLorentz and CPT symmetries should be preserved. Sucharguments, however, do not find support in the history ofphysics. Nearly all known or proposed symmetries, suchas parity and time reversal, electroweak symmetry, chiralsymmetry, and supersymmetry, are spontaneously broken.Whatever the true origin of Lorentz or CPT breaking maybe, the fact that it hasn’t yet been observed means it mustbe small at the energy scales corresponding to knownstandard-model physics.

Effective field theoryHow can one break Lorentz invariance in a controllableway? The least radical approach would be to assume thatlow-energy physics can be described by the Lorentz-invariant dynamics of the standard model plus a numberof possible background fields. Those fields, taken to be con-stant or slowly varying, are vectors or tensors underLorentz transformations and are coupled to ordinary par-ticles in such a way that the whole Lagrangian remainsinvariant. In that framework, called an effective field the-

40 July 2004 Physics Today © 2004 American Institute of Physics, S-0031-9228-0407-020-6

Maxim Pospelov is an associate professor of physics and as-tronomy at the University of Victoria in British Columbia. MichaelRomalis is an assistant professor of physics at Princeton Univer-sity in New Jersey.

Precision experiments and astrophysical observationsprovide complementary tests of Lorentz invariance and maysoon open a window onto new physics. They have alreadyconstrained models of quantum gravity and cosmology.

Maxim Pospelov and Michael Romalis

Lorentz Invariance on Trial

ory, the violation of Lorentz invariance is caused by non-trivial background fields such as the field illustrated in fig-ure 1. The Lorentz violation therefore also appears as aspontaneous symmetry breaking.

One can classify all possible interaction operators bylooking at their dimensions. In the natural units \ ⊂ c ⊂ 1,all interaction operators are energies raised to some posi-tive power called the dimension of the operator. The lowest-dimension operators generally give the largest Lorentz-violating effects. The coefficient of a dimension-D operator inthe Lagrangian has units of energy raised to the power 4⊗D.

Effective-field-theory expansions have been widelyapplied in particle physics since Enrico Fermi, in 1933, pa-rameterized the then-unknown Lagrangian of weak inter-actions by local operators. Effective theories can preserveall of a field theory’s desirable features, including micro-causality, unitarity, gauge invariance, renormalizability,spin statistics, and energy–momentum conservation. Theeffective theory approach to Lorentz violation was advo-cated by Alan Kostelecky (Indiana University) and cowork-ers,3 who developed a systematic generalization of othertheoretical descriptions of Lorentz-violating effects.4–6 Ap-plying the effective-field-theory approach to quantum elec-trodynamics, one finds that the lowest relevant operatordimension is three and that the QED Lagrangian may beexpressed in terms of three dimension-three interactions:

(1)

where c is the electron Dirac spinor, Am is the electromag-netic vector potential, the four-index e is the totally anti-symmetric Levi-Civita tensor, gm are Dirac gamma matri-ces, and g5 and smn are standard combinations of thosegamma matrices. The fields bm, km, and Hmn are external vec-tor and antisymmetric tensor backgrounds that introducea preferred frame and therefore break Lorentz invariance.The first and third terms change sign under the CPT op-eration and therefore violate CPT symmetry.

One can extend the QED Lagrangian by adding otherstandard-model fields or higher-dimension interactions.Operators of dimension five and higher can lead to sig-nificant modifications of the dispersion relations for par-ticles at high energies.7

The effective-field-theory language allowsscientists to systemati-cally study violations ofLorentz invariance. Theycan assign to experimentsa figure of merit that de-pends on the experimen-tal sensitivity to theLorentz-violating effects.The effective-field-theoryframework also providesgeneral guidance as to thetypes of effects one wouldobserve. Among those ef-fects are rotational depen-dences of spin-precessionfrequencies or of the speedof light (such effects canbe measured in low-energy experiments), CPTviolations in neutralmeson systems, and ef-fects that Lorentz viola-

tion would imprint on astrophysical observations.

ComagnetometersModern descendants of Hughes–Drever experiments pro-vide very stringent constraints on many possible Lorentz-violating parameters. Consider the fermionic part (that is,the first two terms) of the Lagrangian given in equation 1.After adding the usual interaction between the magneticfield B and the magnetic dipole moment m(S/S) of a parti-cle with spin S, and then expressing terms in a nonrela-

⊂⇒(3)QED

]xa

]12⊗ ⊗bm

mcg g c5 H k Amnmn

mmnab

ncs c ë⊗ Ab,

http://www.physicstoday.org July 2004 Physics Today 41

b

Sun

Earth

Figure 1. A uniform background vector b defines a preferred direction in space and so violates Lorentz invariance. In effective field theories like those discussed in the text, b isrelated to the three-vector part of a four-vector.

Comagnetometer Measurements

To understand more precisely the Lorentz-violating ef-fects in comagnetometer experiments, which compare

the precession of two particles, consider two nuclei withmagnetic moments m1 and m2. Assume that they couple tothe same Lorentz-violating field b but with different cou-pling coefficients b1 and b2. The precession frequencies ofthe two nuclei are given by

hn1 ⊂ 2m1B ⊕ 2b1(b � nB)

and

hn2 ⊂ 2m2B ⊕ 2b2(b � nB),

where nB is the unit vector in the direction of the spin quan-tization axis defined by the magnetic field B. One can read-ily obtain a combination of the frequencies that is inde-pendent of the external magnetic field:

Thus, in comagnetometer measurements, one can detect aLorentz-violating field only to the extent that it does notcouple to particle spins in proportion to the particles’ mag-netic moments.

Precision measurements of such nonmagnetic spin in-teractions have been used over the years—for example, insearches for a permanent electric dipole moment or forspin-dependent forces mediated by a proposed particlecalled the axion. In single species experiments, one con-cern is the influence of Lorentz-violating effects in com-monly used magnetic shielding; comagnetometer measure-ments, in contrast, are not sensitive to those effects.

n1 n2⊗ ⊗

m1 m1 m2m2( (( (⊂2 b1 b2

( ).b n� Bh

tivistic form, one finds that the Lorentz-violating back-ground couples to spin much like a magnetic field does. In-deed, for S ⊂ 1/2, the interaction Hamiltonian may be ex-pressed as

Hint ⊂ ⊗2mB � S ⊗ 2b � S. (2)

In this equation, b is a vector with components bi ⊗ ëijk

Hjk/2. (The e tensor is totally antisymmetric and we sumover repeated indices.) The spin precession frequency isgiven by

hn ⊂ 2mB ⊕ 2b � nB, (3)

where nB is a unit vector in the direction of the spin quan-tization axis defined by B. Hence, the signal of Lorentz vi-olation would be a change in the precession frequency,which would be caused by rotation of the magnetic fieldrelative to the preferred direction defined by the vector b.Most Hughes–Drever experiments are sensitive to the di-pole coupling between the spin and b vectors. Experimentsusing particles with spin greater than 1/2 or with nonzeroangular momentum are also sensitive to higher-dimensionoperators that have a quadrupole character and induce asignal at twice the magnetic field’s rotation frequency.

The precession frequency depends on the magneticfield, which generally is much greater than a possible bterm. Thus, one must exclude drifts of the magnetic fieldas a possible source of frequency change. Typically, that’sdone with experiments that compare two frequencies pro-portional to the same magnetic field—either the spin pre-cession frequencies of two different species or the spin andorbital precession frequencies of the same species. BecauseLorentz-violating operators coupled to the spin cause onlysmall additive energy shifts, experimentalists generallytry to keep the precession frequency small to maximize the

fractional shift in the frequency.The box on page 41 provides addi-tional details about such meas-urements, usually called comag-netometer or clock-comparisonexperiments.

Most experiments use a mag-netic field fixed on Earth. Theyrely on Earth’s rotation to changethe direction of the field relativeto the preferred frame defined byb and to produce a diurnal modu-lation of the precession frequency.But such a modulation is onlysensitive to the components of bperpendicular to Earth’s rotationaxis. An experiment on a rotatingplatform or in orbit around Earthallows one to constrain other com-ponents of bm. Note that if the com-ponent b0 is nonzero in the refer-ence frame defined by the cosmicmicrowave background, then anexperiment moving with respectto that frame will effectively besubject to a small b field given byb ⊂ (v/c)b0 � 10⊗3 nv b0, where nvis the unit vector in the directionof the experiment’s velocity.

Hughes–Drever experimentsperformed in recent years haveprobed a variety of systems, in-cluding singly charged positiveberyllium ions, atom pairs such as

mercury-199 and mercury-201 or 3He and 21Ne, proton–antiproton or electron–positron pairs in a Penning trap,and muons in a storage ring. With the help of simple atomicand nuclear models, experimenters can interpret theirmeasurements in terms of limits on the Lorentz-violatinginteractions for the electron (be), neutron (bn), and proton(bp). One needs a concrete model for the origin of Lorentz vi-olation, though, to predict the strength of the background-field couplings to different elementary particles.

Laboratory boundsThe best limit on the Lorentz-violating bn term—indeed,the highest overall energy sensitivity to Lorentz-violatingfrequency shifts—has been obtained in a 3He–129Xe maserdeveloped at the Harvard–Smithsonian Center for Astro-physics (CfA).8 Figure 2 illustrates a schematic of the ex-periment. The free spin precession frequencies of the 3Heand 129Xe are simultaneously measured by maintainingpersistent Zeeman maser oscillations for both species. Thetwo atoms operate as a comagnetometer because, absentLorentz violations, their precession frequencies are pro-portional to the same magnetic field. The frequency of the129Xe maser is held constant with the help of a magneticfield adjusted via feedback. Any change in the 3He fre-quency with time would thus indicate an anomalous cou-pling to the nuclear spins.

In a simple nuclear shell model, both 3He and 129Xehave a single valence neutron, and thus their coupling co-efficients, as defined in the box, are equal. But their mag-netic moments differ by a factor of 2.75, and so one retainssensitivity to the neutron Lorentz-violating term. After ap-proximately 90 days of integration, the CfA experiment ob-served, with an uncertainty of 45 nHz, no modulation ofthe 3He precession frequency at Earth’s rotation rate. Suchprecision means that the part of bn perpendicular to Earth’s

42 July 2004 Physics Today http://www.physicstoday.org

Magnetic shields

Pumpbulb

Laserdiodearray795 nm

l/4

B

Pickup coil

Maser bulb

Solenoid

Preamp

I

Detectionelectronics

Currentcontroller

3Hefrequency

129Xe3HeRb

129Xefrequency

Figure 2. A spin maser experiment with helium-3 and xenon-129 atoms sets thebest limit on Lorentz-violating effects for the neutron. A double-bulb glass cellcontains 3He and 129Xe atoms in both chambers and additional rubidium atoms inthe upper bulb. An optical pumping laser array spin-polarizes the Rb atoms,which transfer their polarization to the 3He and 129Xe via spin-exchange collisions.The lower bulb is located inside a pickup coil connected to an external resonatorwith resonances at the Zeeman frequencies for both 3He and 129Xe. Resonancecurrents in the pickup coil maintain a persistent precessing magnetization of bothspin species while the current controller adjusts the magnetic field to maintain a constant spin-precession frequency for the 129Xe. Any change in the 3He spin-precession frequency thus indicates a Lorentz-violating spin interaction. (Figurecourtesy of Ron Walsworth, Harvard–Smithsonian Center for Astrophysics.)

rotation axis has a magnitude less than 5 × 10⊗32 GeV. Thatincredible sensitivity underlines how such bounds can con-strain models of new physics.

The best limit on Lorentz violation for electrons comesfrom a torsion pendulum experiment developed by theEöt–Wash group at the University of Washington. It usesa toroidal pendulum consisting of two different kinds ofpermanent magnets, an aluminum-nickel-cobalt-iron (Al-nico) alloy whose magnetization is mostly produced byelectron spin alignment, and a samarium–cobalt magnetwhose magnetization has a significant contribution fromthe orbital angular momentum of the Sm electrons. By ad-justing the magnetization of the Alnico magnets, theEöt–Wash experimenters can balance the magnetizationin the toroidal ring and almost perfectly cancel the netmagnetic moment of the pendulum. However, because partof the contribution to the magnetization comes from orbitalangular momentum, the pendulum still has a large un-balanced electron spin.

The pendulum hangs from a fiber and rests inside a setof magnetic shields. The whole apparatus sits on a precisionturntable surrounded by Helmholtz coils and suitably posi-tioned masses that reduce magnetic and gravitational fieldgradients. Any Lorentz-violating spin coupling would causea torque on the pendulum; that torque would oscillate at theapproximately 1-hour rotation period of the turntable. Alaser beam reflected from a mirror mounted on the pendu-lum allows the Eöt–Wash researchers to measure the pen-dulum’s rotation with a sensitivity of 4 nanoradians.

The pendulum experiment set limits on the be com-ponents of less than 2 × 10⊗29 GeV. Note that theturntable-mounted experiment constrains the compo-nent of be parallel to Earth’s rotation axis. A similar ex-periment, recently completed at Tsing Hua University inTaiwan, has achieved comparable sensitivity to that of

the Eöt–Wash group.9

Effective field theories canalso include interactions thatallow Lorentz violation in the pho-ton sector: The last term of equa-tion 1 is an example. It and simi-lar higher-dimension terms lead tolight speeds that depend on thepropagation direction or to rota-tions in the polarization of lightpropagating in a vacuum.

Modern descendants of theMichelson–Morley experimentfurnish the best limits on light-speed anisotropies. Stimulated byrenewed interest in tests ofLorentz invariance, groups atStanford University, HumboldtUniversity in Berlin, Germany,and the Observatoire de Parishave recently completed such ex-periments, in which one measuresthe resonance frequency of an op-tical or microwave cavity cooled toliquid He temperatures. As Earthrotates, the orientation of the cav-ity changes relative to a fixed ref-erence frame. If the velocity oflight were to depend on direction,the changing orientation wouldcause a shift in the resonance fre-quency. The most sensitive presentlimits have been set by the Hum-

boldt group, which compared the resonant frequencies oftwo optical cavities made from crystalline sapphire thatwere oriented at 90° to each other.10 They collected datafor about a year and established a limit for variations inthe speed of light of Dc/c � 2 × 10⊗15. Because they col-lected data for a long time and measured signals at vari-ous combinations of Earth’s daily and yearly rotation fre-quencies, the Humboldt researchers were able to placeindependent constraints on nearly all Lorentz-violatingterms that cause anisotropy in the speed of light.

The next generation of experiments is already beingdeveloped. The CfA group is presently working to improvethe long-term stability of their 3He–129Xe maser. They arealso working on improving a hydrogen maser experimentthat has set the best limit on the Lorentz-violating protoninteractions. A group at Amherst College is improving a1995 Hughes–Drever-type experiment11 that compared thespin precession of mercury-199 and cesium-133 atoms andachieved a high short-term sensitivity: Their new experi-ment will sit on a magnetically shielded rotary table. TheEöt–Wash group is also implementing significant im-provements, including an active tilt-stabilization systemfor their turntable and a pendulum with a higher spin mo-ment. A new experiment being developed at Princeton Uni-versity will use potassium and 3He atoms that together op-erate as a self-compensating atomic comagnetometer12

that is sensitive to a particular combination of neutron andelectron violating coefficients, bn/mHe ⊗ be/me. The experi-ment’s principle of operation is illustrated in figure 3.

The experiments now being developed should improveexisting limits by about two orders of magnitude. One chal-lenge faced in experiments designed to look for Lorentz violations is that the slow frequency of signal modulationmakes them very susceptible to 1/f noise. Experimenterscan achieve superior improvements in sensitivity by

http://www.physicstoday.org July 2004 Physics Today 43

a b

Pump PumpP

rob

e

Pro

be

MHe MHe

IHe IHe

SK SK

Bz

Bz

Bz

Bx

B�

B B B� = +z x

Figure 3. A new experiment being developed at Princeton University uses a self-compensating potassium–helium-3 comagnetometer. (a) Optical pumping polar-izes K atoms, which transfer some of their polarization SK to 3He through spin-exchange collisions. The 3He, whose nucleus has spin IHe, develops a significantmagnetization MHe directed opposite to the applied longitudinal magnetic field Bz.The strength of the applied field is adjusted so that the total magnetic field seenby the K atoms vanishes. (b) If the external magnetic field is changing slowly inthe transverse direction, the 3He magnetization adiabatically follows it. As a result, the magnetic field seen by the K atoms remains zero. Absent Lorentz-violating effects, the polarization of K atoms, measured by the probe laser beam,is unchanged. However, a transverse b field that, for example, couples only to theelectron spin of the K atoms will produce a torque that changes the K polarizationwithout affecting 3He. (Figure courtesy of Tom Kornack, Princeton University.)

increasing the signal modulation frequency—either byplacing an experiment on a rotating platform or by plac-ing it on a satellite, as is already planned for futureMichelson–Morley experiments.

CPTParticle physicists have a long history of testing CPT.Since the discovery of antiparticles, increasingly preciseconstraints have been placed on the equality of masses,decay rates, magnetic moments, and other properties ofparticles and antiparticles.

The most stringent constraints on CPT-violatingmass splitting arise from measurements of K0–K0+ mixingand decays. They imply that the CPT-odd parameter+m(K0) ⊗ m(K0+ )+ < 5 × 10⊗19 GeV. Measurements withneutral mesons have the unique feature that they aresensitive to interference effects between particles andantiparticles. Thus, they can constrain model parametersthat cannot be accessed via separate measurements onmatter and antimatter.

Lorentz-symmetry and CPT violations described byeffective field theories should cause many CPT-violatingsignatures in particle physics experiments to acquire si-dereal time dependence that could be exploited to improvesensitivity. In principle, low-energy Hughes–Drever-typeexperiments can achieve good sensitivity to the Lorentz-violating parameters for stable first-generation particles,but experiments with neutral mesons and other unstableparticles are the ones that sensitively probe Lorentz vio-lation for the other two generations.

The coupling of the CPT-odd bm field to particle spinallows one to test CPT without the need for antiparticles:By flipping a particle’s spin, one can measure the same en-ergy shift as would be obtained by comparing a particleand an antiparticle. Thus, g ⊗ 2 experiments and othermeasurements performed with trapped positrons and anti-protons can be directly compared with spin-precessionmeasurements on ordinary particles. If CPT violation isever found, then a comparison of results for particles andantiparticles will allow physicists to separate CPT-oddfrom CPT-even Lorentz-violating effects. They could, forexample, distinguish the separate contributions of bi andëijk Hjk to the b field.

Astrophysical and cosmological testsAstronomical observations can place extremely tight con-straints on certain types of Lorentz violation, such aschanges in the polarization of light propagating throughspace. A rotation in linear polarization is implied, for ex-ample, by the last term of equation 1. For that particularinteraction, the effect is independent of wavelength. Toprobe the effect, one can observe the polarization of syn-chrotron radiation emitted from distant radio galaxies.The initial polarization depends on the emitting galaxy’smagnetic field and can be correlated with the elongationof the galaxy. An analysis of observed synchrotron radia-tion5 constrains the Lorentz-violating coefficient km to havea four-dimensional magnitude less than 10⊗42 GeV. Thatenergy corresponds to a frequency somewhat smaller thanthe Hubble expansion rate.

Other Lorentz-violating terms cause polarization ro-tations that depend on wavelength. They can be measuredby observing visible polarized light emitted by distantgalaxies. Scattering from dust or electrons is typically re-sponsible for polarizing the light, and so the initial polar-ization is independent of wavelength. By measuring thepolarization of scattered light over a range of wavelengths,one can set a limit on wavelength-dependent rotation due

to Lorentz-violating effects.13

Another manifestation of Lorentz violation, which canarise when higher-dimension interactions are included inan effective theory, is a modification of the usual disper-sion relation E2 ⊂ (pc)2 ⊕ (mc2)2. Such modifications leadto a wide variety of effects for high-energy particles—effects that include a spread in the arrival time of photonsfrom gamma-ray bursts;14 limits on the maximum possiblevelocity of a massive particle; and new types of particleprocesses, such as the decay of a high-energy photon intoa positron and electron.4

Observed very high-energy astrophysical processeshave significantly constrained possible modifications tothe standard dispersion relation. For example, synchro-tron radiation from the Crab Nebula (figure 4) has beenobserved with energies of up to 100 MeV, and several meth-ods estimate the nebula’s magnetic field to be in the rangeof 20–50 nanotesla. Together, those two values imply thatelectrons are accelerated up to energies of at least1500 TeV, corresponding to velocities that differ from thespeed of light by less than one part in 1019. The electronsaccelerated to such energies must be stable against thevacuum Ùerenkov radiation process e O ge. Thus, themere existence of such high-energy electrons places strongconstraints on how dispersion relationships for electronsand photons may be modified.15

Challenges for quantum gravityThe Planck mass MPl ⊂ (\c/G)1/2 � 1019 GeV/c2 signals aconceptual problem for quantum field theories. When themomentum transfer in particle collisions is comparable toMPlc, graviton exchange becomes strong and the standardperturbative field-theory description breaks down. Is itpossible that unknown dynamics at the Planck scale couldlead to Lorentz violation? In string theory, a leading can-didate as the theory of quantum gravity, Lorentz- andCPT-violating effects are allowed but not required. Inother approaches to quantum gravity, such as loop quan-tum gravity, it is often argued that the discrete nature ofspacetime at short distances will induce violations ofLorentz invariance and CPT. Such violations lead to dis-persion relations of the form14

(4)

where the ellipses denotes further additions with the gen-eral form En⊕2/(MPlc2)n.

For modified dispersion relations arising from dimension-5 operators, left- and right-handed fermionsmay both be described by the three displayed terms inequation 4, but the j coefficients for the two species areindependent; they are denoted hL and hR, respectively. Asingle j describes the dispersion modification for photons,but with an opposite sign for left- and right-circularly polarized photons.

Astrophysical observations, such as those from theCrab Nebula, constrain various combinations of the elec-tron’s and photon’s j and h parameters with unprece-dented accuracy,15 sometimes to better than 10⊗7. Perhapsmore surprising is that spin precession experiments alsoprovide comparable constraints.7 If one assumes the dis-persion relationships are written in the cosmic microwavebackground rest frame, then any difference between hLand hR for quarks would create an effective spin couplingon the order of 10⊗3 nv (hL ⊗ hR) (mn

2/MPl)c2, where mn isthe nucleon mass. When combined with experimental lim-its from Hughes–Drever experiments, that result yields a

( (⊂E2 ( ) ( )pc mc2 2 2⊕ ⊕ j ⊕ . . .,E3

M cPl2

44 July 2004 Physics Today http://www.physicstoday.org

bound on +hL ⊗ hR+ of about 10⊗8. The electromagnetic in-teractions inside nucleons and nuclei lead to an additionaldependence of spin precession on the j parameter, whosemagnitude can be constrained to be less than 10⊗5. Thus,astrophysical and terrestrial bounds on Lorentz violationfirmly rule out E3/MPl modifications of the dispersion rela-tions for photons, quarks, and electrons—a serious chal-lenge for theories of quantum gravity that predict suchmodifications.

An important and different kind of small-length-scalephysics that leads to the violation of Lorentz invariance isa feature of noncommutative field theories. In such theo-ries, which naturally arise in the context of certain stringtheories,16 spacetime coordinates are noncommuting oper-ators: [xm, xn] ⊂ iqmn. The tensor qmn has the dimensions ofan inverse energy squared and defines a so-called energyscale of noncommutativity LNC � q⊗1/2. At energies belowLNC, noncommuting coordinates would be manifest as in-teractions with standard-model operators of dimension 6such as (qmnFmn)(FabFab), where Fmn is the electromagneticfield tensor.

An antisymmetric tensor, qmn has the same Lorentztransformation properties as the electromagnetic field ten-sor, and so it may be no surprise that its “magnetic” com-ponent qi ⊂ ëijkqjk couples to nuclear spin. The noncommu-tative extension of quantum chromodynamics leads17 to amodulation of the nucleon spin frequency corresponding toan energy of order 0.1 GeV3 L⊗2

NC. That result, combined withHughes–Drever experiments, establishes that the energyscale of noncommutativity is greater than about 1014 GeV.

Considerations of modified dispersion relations andnoncommutative field theories underline the importanceof continued experiments designed to detect Lorentz in-variance. In both examples, the scale of the high-energyphysics responsible for Lorentz breaking can be probedwith unprecedented reach.

Cosmological modelsOur discussion thus far has been in the framework of spe-cial relativity, for which constant Lorentz-violating back-

grounds can be justified. In general relativity, such con-stant fields necessarily become a function of coordinates:Backgrounds such as bm and Hmn acquire kinetic terms andthus participate in dynamics as additional low-energy de-grees of freedom. Although gravitational theories with ad-ditional scalar degrees of freedom have been studied formany years,2 the recent discovery of dark energy has in-tensified research efforts that explore such theories.

A possible candidate for dark energy that avoids someof the fine-tuning problems associated with the cosmolog-ical constant is quintessence, a very low-energy field witha wavelength comparable to the size of the observable uni-verse. In addition to its effect on the expansion of the uni-verse, quintessence might also manifest itself through itspossible interactions with matter and radiation.2,18

One would expect that a quintessence field v relatedto dark energy would still be evolving and that its con-tinuing evolution in time and space would lead to Lorentz-violating effects. As an example, consider the simplestform of scalar and pseudoscalar (that is, containing g5)couplings between a massive Dirac fermion—say, an elec-tron or quark—and the scalar quintessence field v. TheLagrangian

(5)

yields the interaction Hamiltonian

(6)

The scalar interaction leads to a modification of thefermion mass as a function of coordinates and violates the

⊂ ( ).¹v � Smc( )¹v � r ⊕1Fs

1Fa

H int

v v⊗

Fs Fa(⇒ ⊂ ⊗mcc (cig5

http://www.physicstoday.org July 2004 Physics Today 45

1038

1037

1036

1035

1034

1033

10 12 14 16 18 20 22 24LOG FREQUENCY (Hz)

nn

d/d

(erg

/s)

L

10 Gev 10 Tev 100 Tev 3 10 Tev× 3300 Gev

Soft X Hard X GammaOptMicro

Figure 4. (a) X rays in the Crab Nebula are the synchrotronradiation of electrons accelerated in the shock waves cre-ated by the supernova explosion of 1054 AD. This imagewas taken by the Chandra X-ray Observatory. (b) The lumi-nosity L of the synchrotron radiation emitted by the CrabNebula is a function of frequency n. Ground-based obser-vations provide the data for the microwave (Micro) and op-tical (Opt) parts of the spectrum. X-ray and gamma-raymeasurements were performed with detectors aboard theNASA satellites HEAO-1 and the Compton Gamma-RayObservatory. The curve through the data is a theoretical fitfor which the magnetic field in the nebula was taken to be20 nanotesla. Arrows indicate the energy of electronsneeded to produce the various parts of the spectrum: Thehighest-energy electrons severely constrain possible modifi-cations of dispersion relations. (Panel b adapted from A. M.Atoyan, F. A. Aharonian, Mon. Not. R. Astron. Soc. 278,525, 1996.)

a

a

b

equivalence principle: The particle feels an extra force inthe direction of ⊗∇v. The pseudoscalar interaction cou-ples the spin to ∇v and thus creates in the spin precessiona Lorentz-violating signature that is identical to the in-teraction parameterized by b. When a similar derivationis repeated for a photon, one finds that the scalar couplingleads to a fine-structure constant that varies in space andtime and that the pseudoscalar coupling takes essentiallythe same form as the last term in equation 1.

The time and space derivatives of v should be some-what smaller than the square root of the present energydensity in the universe. Assuming that the spatial deriva-tive ∇v has a magnitude on the order of (0.01 eV)2,Hughes–Drever experiments imply that the pseudoscalar(Fa) couplings to electrons and nuclei are greater thanabout 109 GeV. Astrophysical constraints on km imply5,18

that the corresponding pseudoscalar coupling to photonsis greater than the Planck energy.

If future searches for Lorentz invariance and time-dependent fundamental “constants” bring positive results,they may prove that the universe contains a long-wavelengthdegree of freedom and point toward the nature of quintes-sence. Then the variation of fundamental constants andapparently Lorentz-violating spin precessions might becompletely demystified: They could both follow from theconventional physics of an interacting scalar field.

References1. V. W. Hughes, H. G. Robinson, V. Beltran-Lopez, Phys. Rev.

Lett. 4, 342 (1960); R. W. P. Drever, Phil. Mag. 5, 409 (1960).2. C. M. Will, Theory and Experiment in Gravitational Physics,

Cambridge U. Press, New York (1993).3. See, for example, D. Colladay, V. A. Kostelecky, Phys. Rev. D

55, 6760 (1997); Phys. Rev. D 58, 116002 (1998); V. A. Kost-elecky, C. D. Lane, Phys. Rev. D 60, 116010 (1999).

4. S. R. Coleman, S. L. Glashow, Phys. Rev. D 59, 116008 (1999).5. S. M. Carroll, G. B. Field, R. Jackiw, Phys. Rev. D 41, 1231

(1990).6. P. J. Peebles, R. H. Dicke, Phys. Rev. 127, 629 (1962); K. J.

Nordtvedt, Astrophys. J. 161, 1059 (1970); A. P. Lightman,D. L. Lee, Phys. Rev. D 8, 364 (1973); W.-T. Ni, Phys. Rev.Lett. 38, 301 (1977).

7. D. Sudarsky, L. Urrutia, H. Vucetich, Phys. Rev. Lett. 89,231301 (2002); R. C. Myers, M. Pospelov, Phys. Rev. Lett. 90,211601 (2003).

8. D. Bear et al., Phys. Rev. Lett. 85, 5038 (2000); Erratum:Phys. Rev. Lett. 89, 209902(E) (2002).

9. B. R. Heckel et al., (Eöt–Wash group), in CPT and LorentzSymmetry II, V. A. Kostelecky, ed., World Scientific, RiverEdge, NJ (2002); R. Bluhm, V. A. Kostelecky, Phys. Rev. Lett.84, 1381 (2000); L.-S. Hou, W.-T. Ni, Y.-C. M. Li, Phys. Rev.Lett. 90, 201101 (2003).

10. H. Müller et al., Phys. Rev. Lett. 91, 020401 (2003). 11. C. J. Berglund et al., Phys. Rev. Lett. 75, 1879 (1995). 12. T. W. Kornack, M. V. Romalis, Phys. Rev. Lett. 89, 253002

(2002). 13. V. A. Kostelecky, M. Mewes, Phys Rev. Lett. 87, 251304

(2001).14. G. Amelino-Camelia et al., Nature 393, 763 (1998).15. T. Jacobson, S. Liberati, D. Mattingly, Nature 424, 1019

(2003). 16. N. Seiberg, E. Witten, J. High Energy Phys. 1999(09), 032

(1999); M. R. Douglas, N. A. Nekrasov, Rev. Mod. Phys. 73,977 (2001).

17. I. Mocioiu, M. Pospelov, R. Roiban, Phys. Lett. B 489, 390(2000).

18. S. M. Carroll, Phys. Rev. Lett. 81, 3067 (1998). �

46 July 2004 Physics Today